Acoustic resonance in a high-speed axial compressor

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ACOUSTIC RESONANCE IN A HIGH-SPEED AXIAL COMPRESSOR

ACOUSTIC RESONANCE IN A HIGH- SPEED AXIAL COMPRESSOR

Der Fakultät für Maschinenbau

der Gottfried Wilhelm Leibniz Universität Hannover zur Erlangung des akademischen Grades

Doktor-Ingenieur

genehmigte Dissertation von

Dipl.-Phys. Bernd Hellmich

geboren am 23.11.1970 in Rahden/Westfalen, Deutschland

2008

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Referent: Prof. Dr.-Ing. Jörg Seume

Korreferent: Prof. Dr.-Ing. Wolfgang Neise

Vorsitzender: Prof. Dr.-Ing. Peter Nyhuis

Tag der Promotion: 12.11.2007

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„Das Chaos ist aufgebraucht, es war die beste Zeit“

B. Brecht

Für Bettina

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ACOUSTIC RESONANCE IN A HIGH-SPEED AXIAL COMPRESSOR 1. Abstract

1. Abstract

Non-harmonic acoustic resonance was detected in the static pressure and sound signals in a four-stage high-speed axial compressor when the compressor was operating close to the surge limit. The amplitudes of the resonant acoustic mode are in a range comparable to the normally dominating blade passing frequency. This has led to blade cracks in the inlet guided vanes of the compressor where the normal mechanical and aerodynamic load is low.

The present measurements were obtained with a dynamic four-hole pneumatic probe and an array of dynamic pressure transducers in the compressor casing. For signal decomposition and analysis of signal components with high signal-to-noise ratio, estimator functions such as Auto Power Spectral Density and Cross Power Spectral Density were used.

Based on measurements of the resonance frequency and the axial and circumferential phase shift of the pressure signal during resonance, it is shown that the acoustic resonance is an axial standing wave of a spinning acoustic mode with three periods around the circumference of the compressor. This phenomenon occurs only if the aerodynamic load in the compressor is high, because the mode needs a relative low axial Mach number at a high rotor speed for resonance conditions. The low Mach number is needed to fit the axial wave length of the acoustic mode to the axial spacing of the rotors in the compressor. The high rotor speed is needed to satisfy the reflection conditions at the rotor blades needed for the acoustic resonance.

The present work provides suitable, physically based simplifications of the existing mathematical models which are applicable for modes with circumferential wavelengths of more than two blade pitches and resonance frequencies considerably higher than the rotor speed. The reflection and transmission of the acoustic waves at the blade rows is treated with a qualitative model. Reflection and transmission coefficients are calculated for certain angles of attack only, but qualitative results are shown for the modes of interest. Behind the rotor rows the transmission is high while in front of the rotor rows the reflection is dominating. Because the modes are trapped in each stage of the compressor by the rotor rows acoustic resonance like this could appear in multi stage axial machines only.

Different actions taken on the compressor to shift the stability limit to lower mass flow like dihedral blades and air injection at the rotor tips did not succeed. Hence, it is assumed that the acoustic resonance is dominating the inception of rotating stall at high rotor speeds. A modal wave as rotating stall pre cursor was detected with a frequency related to the acoustic resonance frequency. In addition the acoustic resonance is modulated in amplitude by this modal wave. The acoustic waves are propagating nearly perpendicular to the mean flow, so that they are causing temporally an extra incidence of the mean flow on the highly loaded blades, but it could not be proved if this triggers the stall. A positive effect of the acoustic resonance on the stability limit due to the stabilization of the boundary layers on the blades by the

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ACOUSTIC RESONANCE IN A HIGH-SPEED AXIAL COMPRESSOR 1. Abstract

Kurzfassung

Nicht drehzahlharmonische akustische Resonanzen sind in den statischen Drucksignalen in einem vierstufigen Hochgeschwindigkeitsaxialverdichter gemessen worden, als dieser nahe der Pumpgrenze betrieben worden ist. Die Amplituden der resonanten Moden lagen in derselben Größenordnung wie die Schaufelpassier- frequenz. Das führte zu Rissen in den Schaufeln im Vorleitapparat des Kompressors, wo die normale mechanisch und aerodynamisch Belastung niedrig ist.

Die vorliegenden Messungen sind mit einer dynamischen Vierloch-Sonde und einem Feld von dynamischen Drucksensoren im Kompressorgehäuse durchgeführt worden.

Zur Signalzerlegung und Analyse mit hohem Signal zu Rauschverhältnis sind Schätzfunktionen wie Autospektrale Leistungsdichte und Kreuzspektrale Leistungsdichte verwendet worden.

Aufgrund von Messungen der Resonanzfrequenz sowie des axialen und peripheren Phasenversatzes der Drucksignale unter resonanten Bedingungen ist gezeigt worden, dass die akustische Resonanz in axialer Richtung eine stehende Welle ist mit drei Perioden um den Umfang. Das Phänomen tritt nur auf, wenn die aerodynamische Belastung des Kompressors hoch ist, da die Resonanzbedingungen für die Mode eine niedrige axiale Machzahl bei hoher Rotordrehzahl erfordern. Die relativ niedrige axiale Machzahl ist nötig, damit die axiale Wellenlänge zum axialen Abstand der Rotoren passt. Die hohe Drehzahl ist notwendig, um die Reflektionsbedingungen an den Rotorschaufeln zu erfüllen.

Die vorliegende Arbeit verwendet physikalisch basierte Vereinfachungen von existierenden Modellen, die anwendbar sind auf Moden mit einer Wellenlänge in Umfangsrichtung, die größer als zwei Schaufelteilungen sind und eine Resonanzfrequenz haben, die über der Rotordrehzahl liegt. Die Reflektion und Transmission von akustischen Wellen ist mit einem qualitativen Modell behandelt worden. Die Reflektions- und Transmissionskoeffizienten sind nur für bestimmte Einfallswinkel berechnet worden, aber die Ergebnisse zeigen, dass für die relevanten Moden die Transmission hinter den Rotoren hoch ist während vor den Rotoren die Reflexion dominiert. Weil die Moden zwischen den Rotoren eingeschlossen sind, können akustische Resonanzen wie diese nur in mehrstufigen Axialmaschinen auftreten.

Verschiedene Maßnahmen, die an dem Kompressor durchgeführt worden sind, wie dreidimensionale Schaufelgeometrien oder Einblasung an den Rotorspitzen, haben zu keiner Verschiebung der Stabilitätsgrenze zu niedrigeren Massenströmen geführt.

Deshalb wird angenommen, dass die akustische Resonanz die Entstehung einer rotierenden Ablösung (Rotating Stall) herbeiführt. Eine Modalwelle als Vorzeichen für den Rotating Stall ist gemessen worden, dessen Frequenz ein ganzzahliger Bruchteil der Frequenz der akustischen Resonanz ist. Außerdem ist die Amplitude der akustischen Resonanz von der Modalwelle moduliert. Die akustischen Wellen laufen nahezu senkrecht zur Strömungsrichtung, so dass sie kurzzeitig eine zusätzliche Fehlanströmung der hoch belasteten Schaufeln verursachen, aber es konnte nicht bewiesen werden, dass dies die Ablösung auslöst. Ein positiver Einfluss der akustischen Resonanz auf die Stabilität des Verdichters durch die Stabilisierung der Grenzschichten auf den Verdichterschaufeln ist ebenso nicht ausgeschlossen.

Schlagwörter: Akustische Resonanz, Axialverdichter, Rotierende Ablösung

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ACOUSTIC RESONANCE IN A HIGH-SPEED AXIAL COMPRESSOR 2.Contents

2. Contents

1. Abstract...V 2. Contents...VII 3. Nomenclature ...IX

4. Introduction ... 1

4.1. Compressors ... 1

4.2. Flow distortions in compressors ... 3

5. Literature Review ... 5

6. Test facility... 8

6.1. The test rig ... 8

6.2. Sensors, Signal Conditioning and Data Acquisition... 10

7. Signal processing methods ... 12

7.1. Auto Power Spectral Density (APSD)... 12

7.2. APSD estimation by periodogram averaging method... 13

7.3. Cross power spectral density (CPSD), phase, and coherence... 15

7.4. Coherent transfer functions ... 17

7.5. Statistical Errors of coherence, power and phase spectra ... 18

7.6. Spectral leakage... 20

8. The Phenomenon ... 24

8.1. Acoustic resonance versus multi-cell rotating stall ... 27

8.2. Helmholtz resonance at outlet throttle... 28

8.3. Vibration induced blade cracks ... 30

9. Theoretical Prediction of resonant modes... 33

9.1. Effect of blade rows... 33

9.2. Effect of non-uniform flow and geometry... 37

9.3. Summary of theoretical prediction methods ... 38

9.4. Simplified model ... 38

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ACOUSTIC RESONANCE IN A HIGH-SPEED AXIAL COMPRESSOR 2.Contents

10. Application to Measurements ... 45

10.1. Measured pressure magnitude and phase shifts... 48

10.2. Measured radial pressure distribution of AR... 48

10.3. Application of model to flow measurements... 52

10.4. Reflection and transmission of acoustic waves through blade rows ... 58

10.5. Transient flow measurements at constant rotor speed... 62

10.5.1. Frequency shift and mean flow parameters...63

10.5.2. Axial phase shift and mean flow parameters...64

10.6. Acoustic resonance and rotor speed... 67

10.7. Summary and conclusions of data analysis ... 71

11. Rotating Stall inception... 72

11.1. Compressor stability... 72

11.2. Fundamentals of rotating stall ... 73

11.3. Stall inception in the TFD compressor... 77

11.3.1. Diffusion coefficient ...77

11.3.2. Axial detection of the rotating stall origin ...78

11.3.3. Radial detection of the rotating stall origin...82

11.3.4. Circumferential detection of the rotating stall origin ...83

11.3.5. Stall inception at different rotor speeds ...84

11.4. Frequency analysis of rotating stall measurements... 85

11.4.1. Magnitudes at different rotor speeds...85

11.4.2. Phase and Coherence at different rotor speeds ...86

11.5. Summary of rotating stall inception ... 91

11.6. Interaction of acoustic resonance and rotating stall ... 91

11.6.1. Time series analysis of measurements ...93

11.6.2. Modulation of acoustic resonance ...94

11.6.3. Magnitude and incidence angle of acoustic waves prior stall ...95

11.7. Conclusions of acoustic resonance and rotating stall interaction ... 97

12. Summary and Conclusions ... 98

13. Outlook... 100

14. Acknowledgement... 103

15. References... 104

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ACOUSTIC RESONANCE IN A HIGH-SPEED AXIAL COMPRESSOR 3.Nomenclature

3. Nomenclature

Symbol Units Meaning Defined in

Latin

a m/s speed of sound

c m/s mean flow velocity

c_u / c_a m/s swirl / axial velocity of mean flow d m diameter of the compressor annulus

D m tip clearance, damping factor Eq. (54) Section 10.3

DC diffusion coefficient Eq. (64) Section 11.3.1

f Hz absolute frequency

f_shaft Hz shaft speed

k 1/m wave number Eq. (41) Section 9.4

l m chord length

Ma Mach number of the mean flow

Ma_φ /Ma_z circumferential / axial Mach number of the mean flow

m&corr kg/s normalized mass flow

m circumferential mode number

n Hz physical rotor speed, radial mode number

nnorm Hz normalized rotor speed

p Pa pressure

r m radius

s m blade pitch

t s time

z m axial co-ordinate

h Index of the harmonic order

Greek

α ° flow angle Eq. (48) Section 9.4

αs ° stagger angle of the blade row α_m ° blade inlet/exit angle

β_± ° incidence angle of wave in present model, slope angle of helical wave fronts

Eq.(45) Section 9.4

λ m wave length

θI ° incidence angle of wave in Koch’s model

Fig. 17 θ_R ° reflection angle of wave in Koch’s

model

Fig. 17 θT ° transmission angle of wave in Koch’s

model

Fig. 17

π total pressure ratio

σ hub ratio (r_i/ r_o)

φ ° azimuthal coordinate

ω 2π/s angular frequency

δ ^° circumferential distance of sensor x

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ACOUSTIC RESONANCE IN A HIGH-SPEED AXIAL COMPRESSOR 4.Introduction

4. Introduction

4.1. Compressors

“My invention consists in a compressor or pump of the turbine type operating by the motion of sets of movable blades or vanes between sets of fixed blades, the movable blades being more widely spaced than in my steam turbine, and constructed with curved surfaces on the delivery side, and set at a suitable angle to the axis of rotation. The fixed blades may have a similar configuration and be similarly arranged on the containing casing at any suitable angle. “Parsons 1901, taken from Horlock (1958)

In 1853 the basic fundamentals of the operations of a multistage axial compressor were first presented to the French Academy of Sciences. Parsons built and patented the first axial flow compressor in 1901 (Horlock (1958)). Since then, compressors have significantly evolved. For example, continuous improvements have enabled increases in efficiency, the pressure ratio per stage, and a decrease in weight. Compressors have a wide variety of applications. They are a primary component in turbojet engines used in aerospace propulsion, in industrial gas turbines that generate power, and in processors in the chemical industry to pressurize gas or fluids. The size ranges from a few centimetres in turbochargers to several meters in diameter in heavy-duty industrial gas turbines. In turbomachinery applications, safe and efficient operation of the compression system is imperative. To ensure this and to prevent damage, flow instabilities must be avoided or dealt with soon after their inception. Considerable interest exists in the jet propulsion community in understanding and controlling flow instabilities. Especially turbojet engines in military aircraft and gas turbines in power plants during instabilities of the electric grid must be able to handle abrupt changes in operating conditions. Hence, the treatment of flow instabilities in axial flow compressors is of special interest in military applications and power generation.

1.1 AN OVERVIEW OF COMPRESSOR OPERATIONS

The basic purpose of a compressor is to increase the total pressure of the working fluid using shaft work. Depending on their type, compressors increase the pressure in different ways. They can be divided into four general groups: rotary, reciprocating, centrifugal and axial. In reciprocating compressors, shaft work is used to reduce the volume of gas and increase the gas pressure. In rotary compressors gas is drawn in through an inlet port in the casing, captured in a cavity and then discharged through

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FIG. 1 SCHEMATIC DIAGRAM OF CHANGES IN FLUID PROPERTIES AND

VELOCITY THROUGH AN AXIAL COMPRESSOR STAGE

high kinetic energy of the fluid is converted into pressure by decelerating the gas in stator blade passages or in a diffuser. In centrifugal compressors, the flow leaves the compressor in a direction perpendicular to the rotation axis. In axial compressors, flow enters and leaves the compressor in the axial direction. Because an axial compressor does not benefit from the increase in radius that occurs in a centrifugal compressor, the pressure rise obtained from a single axial stage is lower. However, compared to centrifugal compressors, axial compressors can handle a higher mass flow rate for the same frontal area. This is one of the reasons why axial compressors have been used more in aircraft jet engines, where the frontal area plays an important role. Another advantage of axial compressors is that multistaging is much easier and does not need the complex return channels required in multiple centrifugal stages. A variety of turbomachines and positive displacement machines as well as their ranges of utilization in terms of basic non-dimensional parameters are shown in Fig. 2. The horizontal axis represents the flow coefficient, which is a non-dimensional volume flow rate. The vertical axis shows the head coefficient, which is a dimensionless measure of the total enthalpy change through the stage, and roughly equals the work input per unit mass flow.

The compressor considered in this study is an axial compressor. As shown in Fig. 1, it consists of a row of rotor blades followed by a row of stator blades. The working fluid passes through these blades without any significant change in radius. Energy is transferred to the fluid by changing its swirl, or tangential velocity, through the stage. A schematic diagram of the changes in velocity and fluid properties through an axial compressor stage is shown in the lower diagram of Fig. 1 taken from references Gravdahl (1999) and Japikse and Baines (1994). It shows how pressure rises through the rotor and stator passages.

Early axial compressors had entirely subsonic flow. Since modern applications require compression systems with higher pressure ratios and mass flow rates, designers have permitted supersonic flow, particularly near the leading edge tip where the highest total velocity occurs.

Today, most high performance

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compression stages are transonic, where regions of subsonic and supersonic flow both exist in the blade passages. The steady state performance of a compressor is usually described by a plot of the mass flow rate versus the total pressure ratio. This plot is called the characteristic or performance map of the compressor. The performance map of the four stage compressor used for this work is shown in Section 6.1.

4.2. Flow distortions in compressors

Beside the well known phenomenon of rotating stall and surge, the most recent literature reports other phenomena of flow distortion or even flow instability in compressors, called rotating instabilities (März et al. (2002), Weidenfeller and Lawerenz (2002), Mailach et al. (2001), Kameier and Neise (1997), Baumgartner et al.

(1995)), tip clearance noise (Kameier and Neise (1997)), and acoustic resonance (AR).

As will be described more in detail in Chapter 11, rotating stall is a rotating blockage propagating by flow separation in neighbouring channels due to an incidence caused by the blocked channel. Rotating instabilities are interpreted as periodic flow separations on the rotor without blockage. They appear as wide flat peaks in the frequency spectra of pressure signals of pneumatic probes or wall-mounted pressure transducers at a frequency higher than the rotor speed. Wide flat peaks are caused by pressure fluctuations with high damping. The damping is caused by the wash out of the flow separations in the down stream regions of the rotor. The tip-clearance noise looks similar, but it is caused by reversed flow at the rotor tips and is strongly dependent on the tip clearance gap. Compared to these effects, acoustic resonances cause a high and narrow peak in the pressure signal spectrum at a certain resonance frequency.

This means compared to a rotating instability the damping is low. Because rotating instability and acoustic resonance may occur in the same frequency range, it is not excluded that a rotating instability works as a driving force for an acoustic resonance.

Unlike the effects mentioned above, acoustic resonances themselves must be explained by an acoustic approach. A simple example of an acoustic resonator is an organ whistle where the tube is a resonator driven by broadband noise caused by shear layers in the flow. The driving forces of a resonance could be aerodynamic effects, like vortex shedding, rotor-stator interaction or shear layers in the flow for example. In turbomachinery, the resonance conditions are typically satisfied only at specific flow conditions.

As will be shown in the next chapter, the relevance of these effects in modern turbomachinery is still under discussion.

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FIG. 2 WORK INPUT MACHINERY CLASSIFICATION (JAPIKSE AND BAINES (1994))

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ACOUSTIC RESONANCE IN A HIGH-SPEED AXIAL COMPRESSOR 5.Literature Review

5. Literature Review

In the beginning the main purpose of aero-acoustics was the noise reduction of turbomachinery. Due to the wide use of jet engines in civil aircraft, noise reduction became necessary to avoid high noise emission in the airport areas and inside the planes. After the noise emissions of the trust had been reduced through higher bypass ratios, compressor and fan noise emissions became more relevant. Blade passing frequencies and rotor-stator interaction have been found to be the major noise source of fans and compressor stages. Tyler and Sofrin (1962) and Lighthill (1997) did pioneering work in this field.

Their papers dealt with jet noise and rotor-stator interaction which always occurs at blade-passage frequency or an integer multiple of it with a phase velocity that is a multiple of the rotor speed. By contrast, acoustic resonances are characterised by discrete frequencies which are in general not an integer multiple of rotor speed. Here pioneering work has been done by Parker and co-workers (Parker (1966), Parker (1967), Parker (1984), Parker and Stoneman (1987), Parker and Stoneman (1989), and Legerton et al. (1991)). For instance, in an experiment with plates in a wind tunnel Parker could show that the resonance occurs slightly above the air-speed for which the natural shedding frequency equals the resonance frequency. This frequency was also predicted by solving the acoustic wave equations for the wind tunnel and the plate. The resonances Parker found are standing waves within the blade pitch or between blades and the tunnel wall.

In a similar experiment, Cumpsty and Whitehead (1971) found that the presence of an acoustic field was correlated with the eddy shedding by the plate over the whole span of the plate, while it was correlated over short length scales at off-resonance conditions only.

Compared to the wind tunnel, the conditions in a turbomachine are far more complicated. With regard to this, the transfer of Parker’s results to a real machine is fraught with two major problems: Firstly, as will be shown below, the acoustic modes of an annular duct with non-uniform swirl flow, varying cross section, and blade rows are rather complicated. Secondly, the blades in turbomachines are heavily loaded at off- design conditions and the flow speed is high compared to those in Parker’s experiments. In view of this, both the acoustics and aerodynamics are more complicated than in the case of Parkers wind tunnel experiments. This does not mean that acoustic resonances could not exist in turbomachinery, but it makes it hard to predict them in advance. Nevertheless, if a machine exhibits resonant behaviour under certain flow conditions, it is sometimes possible to simplify the problem and to show that the anomalous behaviour fits an acoustic resonance.

Before different mathematical approaches for sound propagation through ducts and

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In one of his early papers, Parker refers to the findings of Rizk and Seymour (1964) who reported an acoustic resonance driven by vortex shedding in a gas circulator at Hinkley Point Nuclear Power Station in the UK.

After the above-mentioned study by Parker, some experimental and theoretical papers on acoustic resonances in turbomachines followed in the 1970s and 1980s. The next acoustic resonance from a real machine reported in the literature to the knowledge of the author is mentioned by Von Heesen (1997). He was facing the problem that the same type of blowers was noisy in some street tunnels, while in others they where not.

He found that the noisy ones were running at a different operating point and with a different axial velocity of the flow. As a cause, he found that the frequency of a helical acoustic mode in the duct fitted the vortex shedding frequency of the blades. He suppressed this mode by extending a few stator vanes in axial direction, so that the propagation of helical modes in circumferential direction was suppressed.

Later Ulbricht (2001) found an acoustic mode in a ring cascade. The circumferential phase velocity of the acoustic field she measured was above the speed of sound. This is typical for helical modes as explained in Section 9.4.

In a similar ring cascade, Weidenfeller and Lawerenz (2002) found a helical acoustic mode at a Mach number of 0.482. They showed that the acoustic field was present at the hub and tip of the blades and measured significant mechanical blade stresses at the frequency of the acoustic mode. In a similar way, Kameier (2001) found some unexpected unsteady pressure fluctuations when he tested the aircraft engine of the Rolls Royce BR 710 and BR 715.

Camp (1999) carried out some experiments on the C106 low-speed high-pressure axial compressor and found a resonance quite similar to the acoustic resonance that is the subject of to this work and previous papers of the present author (Hellmich and Seume (2004) and (2006)), i.e. he found a helical acoustic mode. Camp further assumed vortex shedding of the blades as the excitation mechanism. He found Strouhal numbers around 0.8 based on the blade thickness at the trailing edge. He argued that this is far above the usual value of 0.21 for vortex shedding of a cylinder and suggested that this was due to the high loading of the blades. As a consequence, the effective aerodynamic thickness of the blades was higher than the geometric thickness.

Ziada et al. (2002) reported an acoustic resonance in the inlet of a 35 MW Sulzer multistage radial compressor in a natural gas storage station in Canada. The resonance had lead to mechanical vibration levels of the whole compressor resulting in a vibration at the compressor casing above the specified limit of vibration for this class of machines. The resonance was driven by vortex shedding struts and could be eliminated by a modification of the struts’ trailing edges. This effect is also well known for tube bundles in heat exchangers.

In a study supported by General Electric, Kielb (2003) found non-synchronous vibrations in a high-speed multi-stage axial-compressor. The size of the compressor was not specified in the paper, but the resonance frequency he found was in the same range as those measured at the rig in this work.

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Vignau-Tuquet and Girardeau (2005) analyzed non-engine order pressure fluctuations with discrete frequencies between rotor speed and blade passing frequency (BPF) in a three-stage high-speed compressor test rig. As they measured in the fixed and rotating frame, they were able to make a clear estimation of the circumferential modes of the pressure fluctuation. The mode numbers and rotating speed of the modes fitted an acoustic resonance.

A recent paper by Cyrus et al. (2005) deals with a phenomenon that might be an acoustic resonance. They faced the problem of stator vane damage in the rear stages of an Alstom gas turbine compressor. With pneumatic pressure probes they found that a non-synchronous pressure oscillation with a frequency close to a natural frequency of the damaged blades existed in the rear stages of the compressor. From the measurements and numerical calculations it turned out, that “stalled flow on vane surfaces with vortex flow shedding” (Cyrus et al. (2005)) was responsible for the flow pulsations.

All these cases can be summarized by the following features:

• Acoustic resonances occur in real machines and cascades inside an annulus.

• They occur as non-synchronous pressure fluctuations at discrete frequencies.

• The acoustic field in most cases has a helical structure.

• In most cases vortex shedding was assumed to be the excitation of the resonance.

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ACOUSTIC RESONANCE IN A HIGH-SPEED AXIAL COMPRESSOR 6.Test facility

6. Test facility

The acoustic resonance of the Hannover four-stage high-speed axial compressor has already been discussed in previous papers by Hellmich and Seume (2004) and (2006).

However, for a better understanding the results will be summarized in the following section.

6.1. The test rig

A detailed description of the test rig with performance map is given by Fischer et al.

(2003), Walkenhorst and Riess (2000), and Braun and Seume (2006). The measurements provided here are measured with the configuration they refer to as the

“reference” configuration in their work.

4-hole dynamic pneumatic probe location

pressure transducers in the casing

outlet throttle Blade numbers rotor

23 27 29 31

Ø400 1150

Ø500

26 30 32 34 36 Blade numbers stator

FIG. 3 TURBOMACHINERY LABORATORY FOUR STAGE HIGH SPEED AXIAL COMPRESSOR

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max. rotor speed nmax 300 Hz corrected mass flow with

preference = 60000 Pa ^corr

m& 8.3 kg/s

inlet total pressure p_e⁰ 0.6 × 10⁵ Pa total isentropic efficiency ηis _{88,6 %}

total pressure ratio π 2.98

number of stages z 4

stage pressure ratio π_St 1.3

outer radius r_o 170 mm

blade height h 90 – 45 mm

axial velocity c_z 190 – 150 m/s

circumferential velocity cc max. 320 m/s tip clearance gap (cold) s 0.4 mm number of rotor blades N_R 23, 27, 29, 31

number of stator vanes N_S 26(IGV), 30, 32, 34, 36

TABLE 1 COMPRESSOR DESIGN DATA

6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 2.00

2.25 2.50 2.75 3.00 3.25 3.50

0.70 0.75 0.80 0.85 0.90 0.95 1.00 nr/n

max =0.95

pressure ratioππππ

corrected mass flow [kg/s]

static pressure outlet/total pressure inlet static pressure ratio

total pressure ratio

isentropic efficency

isentropic efficency

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6.2. Sensors, Signal Conditioning and Data Acquisition

The rotor speed, the position of the throttle and the ten pressure signals (c.f. Table 2 for names and positions) were recorded synchronously with a high-speed data acquisition system. The reduced mass flow and pressure ratio were computed by the compressor operating software and the compressor standard measuring system. The rotor speed was provided by one TTL pulse per revolution. The throttle position was measured by a potentiometer that was connected to a mechanical position indicator of the throttle. The unsteady pressure signal was provided by Kulite XCQ-062 piezo- resistive pressure transducers with a resonance frequency of 300 kHz and a pressure range from –100 to 170 kPa. The transducers had been calibrated against static reference pressure before the measurements were performed. The transducer signal was a voltage difference which was amplified and transformed to single-ended signals.

FIG. 5 ROLL UP OF STATOR BLADES WITH SENSOR POSITIONS

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Pneumatic Probe (Kulite XCQ-062)

Total pressure dto

Radial Postion [% of channel height]

Axial Position

2 5 8 11 14

Behind Rotor 1 95.2 % 86.8 % 50.8 % 15.7 % 5.0 %

Behind Rotor 2 87.8 % 66.2 % 45.9 % 25.7 % 6.8 %

Pressure transducers in the casing (Kulite XCQ-062) Axial Position in front of Circumferencial position

Rotor 1 Rotor 2 Rotor 3 Rotor 4 Outlet

0° d11 d13 d15 d17 d19

71° d21

255° d41 d43 d45 d47 d49

301° d51

TABLE 2 SENSOR POSITIONS AND NOMENCLATURE

All unsteady signals were measured by sigma-delta analog-to-digital converters (ADCs) with an input voltage range of ±10 V and 16 Bit resolution. This provided a pressure resolution better than 5 Pa. The ADCs were mounted on a Chico+™ PCI Baseboard produced by Innovative DSP Inc ™. As data logger, an Intel Pentium based server was used. The ADCs had integrated anti-aliasing low-pass filter with a frequency bandwidth of 50% of the sampling rate. The maximum sampling rate was 200 kHz for each channel. For the present experiments, 16 Channels with a sampling rate between 20 kHz and 60 kHz were used.

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ACOUSTIC RESONANCE IN A HIGH-SPEED AXIAL COMPRESSOR 7.Signal processing methods

7. Signal processing methods

7.1. Auto Power Spectral Density (APSD)

Unsteady pressure signals of probes and transducers mounted in turbomachinery usually are a mixture of stochastic and ergodic signal components. The stochastic parts of the signal are background noise of the measuring line and pressure fluctuation due to shear layers in the flow. The ergodic signal components arise from the pressure patterns that are rotating with the rotor of the machine. As to this, they occur as periodic pressure fluctuations in the fixed frame of the flow. The common example therefore is the blade passing frequency (BPF). Beside this, in some cases periodic pressure fluctuations occur which have non engine-order frequencies. Common examples are rotating stall, surge, modal waves, and acoustic modes. Usually, the blade passing frequency and its higher orders are dominating the signal. Other periodic components are often hidden in the background noise since their magnitude is in the same range or even lower than the stochastic background noise.

If a signal contains more than one periodic component, it can be decomposed by a transformation of the time series signal in the frequency domain.

A rather powerful method to find hidden periodic signal components in the background noise of the raw signal in the frequency domain are stochastic estimator functions like the Auto Power Spectral Density (APSD) function. This is the Auto Correlation Function (ACF) in the frequency domain. The ACF of a data set of real values x(t) of length T is defined as

( )

⁼

∫

^T

( ) (

⁻

)

xx xt x t dt

ACF T

0

1 τ

τ (1)

A straight forward method to estimate the APSD is the Fourier transformation of the ACF.

( )

⁼

∫

^T ^xx

( ) (

⁻

)

XX f ACF if d

APSD

0

2

exp π τ τ

τ (2)

However, in practice where time series data are given by discrete values, this method produces a high variance in the estimation of periodograms. This could be improved by averaging in the frequency domain. The common method to estimate the APSD with a low variance is described in the next subsection.

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7.2. APSD estimation by periodogram averaging method

The periodogram averaging method provides an estimation of the APSD where the variance is reduced by a factor equal to the number of periodograms considered in the averaging (see Bendat and Piersol (2000) for details). Therefore the original sequence is partitioned into smaller non-overlapping segments (or 50% overlapping in some methods), and the periodograms of the individual segments are computed and averaged. Unfortunately, by reducing the number of samples in each sub-sequence through segmentation, the frequency resolution of the estimate is simultaneously reduced by the same factor. This means that for a sequence of a given length there is a trade off between frequency resolution and variance reduction. High frequency resolution and low variance is possible to achieve by using a long data sequence measured under stationary conditions only.

The procedure for generating a periodogram average is as follows:

Partition the original sequence x[m] into multiple segments of equal/shorter length by applying a window function

[ ]

ⁿ ^x

[

ⁿ ^kM

] [ ]

^wⁿ

x_k = + ⋅ (3)

where the window length is M and w[n] a window function that forces the function x_k[n]

to be periodic.

With a Fourier transformation, periodograms of each segment are calculated:

[ ] [ ] [ ]

1 2

0

2 2

1

∑

⁻ exp

=



 



−

=

M

m k j

k j k

XX M

i jm m

M x f

X f

S π

and f_j=jf_s/M (4)

with j=0,1,2,…,M-1 and fs as sampling rate of the data acquisition. The assumptions of wide sense stationarity and ergodicity allow us to apply the statistics of the segment to the greater random sequence and random process respectively.

Averaging the segmented periodograms estimates together the APSD:

[ ] [ ] ∑

⁻

[ ]

=

1

0

1 ^K

k

j k XX j

XX j

XX S f

f K S f

APSD ₍₅₎

where K= N/M is the number of segments.

It should be noted that in some cases other procedures for PSD estimation are more efficient like Auto Regressive Modeling or Multi Taper Method.

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The unit of the APSD is [x]²/Hz if [x] is the unit of x(t) and the time is measured in seconds. For practical issues this unit is not always useful. Because of this, in many cases the magnitude function Mag is used. It is given by:

[ ]

_k

XX

k APSD f

f M

Mag 2

]

[ = (6)

Its unit is the same as x(t).

Due to Parsevals theorem, the integral of a function f(t) over all times equals the integral over its Fourier transformed function F(f) over all frequencies. This makes it possible to normalize the APSD to the variance of a time signal used for the APSD estimation. The variance is given by:

( )

(

^x^t ^T

)

⁼_T

∫

^T

(

^x

( )

^t ⁻^x

)

^dt

Var

0

1 2

, (7)

In this work all APSDs are normalized so that the following equation is fulfilled:

[ ]

( ) ( [ ] ) [ ]

( )

∑

⁻

=

−

=

−

=

1 2 /

0 1

0

1 2

,

M

m

m XX N

n

f APSD x

n N x

N n x

Var ₍₈₎

Because the APSD is an even function it must be summed up only for the positive frequencies. For the negative frequencies the sum is the same so that the sum over all frequencies is two times the sum over the positive or the negative frequencies. The common normalization is given by:

[ ]

( ) ( [ ] ) [ ]

( )

∑

⁻

=

−

=

∆

=

−

=

1 2 /

0 1

0

1 2

,

M

m

m XX N

n

f APSD f

x n N x

N n x

Var with

M f = f^s

∆ (9)

The normalization used here is uncommon and already includes the normalization to the frequency resolution ∆f. So the APSD used here is in reality not a density anymore.

In correct words it should be called Auto Power Spectrum (APS), but this is not done here. The signal processing tools used for the processing of the data in the next chapters are using this uncommon normalization and hence it is used here, too. In this convention the standard deviation of a signal is given by:

[ ]

( ) ( [ ] ) [ ]

( )

∑

⁻

=

1 2 /

0

, ,

. .

M

m

m XX f APSD N

n x Var N

n x d

s (10)

If we assume a time signal xf[n] filtered with a narrow band pass filter of band width

∆f=f_s/M around the center frequency of f_m, the standard deviation of this signal would be

(25)

(

^x_f

[ ]

ⁿ ^N

)

^APSD_XX

[ ]

^f_m

d

s. . , = (11)

In other words, the normalization makes it possible to use the effective oscillation amplitude given by the standard deviation, which in turn could be derived directly from the APSD. However, this only applies to cases where the peak caused by the oscillation in the spectra spreads over only one frequency slot. In reality, this is usually not the case. With regard to this, a correct estimation would require the summation of the APSD over the width of the peak, which varies from peak to peak. So, as a compromise we introduce a magnitude function using a moving window for summing up the amplitude at the center frequency together with the amplitudes of the neighboring frequency slots.

[ ] ∑

⁺

[ ]

−

=

1

1 m

m i

i XX m

XX f APSD f

Mag (12)

So the peak value of a wide peak in a magnitude spectra is an over prediction of the effective oscillation amplitude of the associated oscillation, because the root of a sum of values is always less than the sum of the roots of the single values.

A special case is the harmonic analysis. Here, the averaging is done in the time domain and the FFT is performed from the averaged data. In this case the magnitude is estimated directly from the FFT with

[ ]

2 Re

( [ ] )

² Im

( [ ] )

²

, _m _m

m

XX X f X f

N N f

Mag = + (13)

where X is the Fourier transformation of x and N is the length of the window used for the discrete Fourier transformation.

7.3. Cross power spectral density (CPSD), phase, and coherence

Analog to the calculation of the power spectral density of a signal with it self, the power spectral density of two different time series x[n] and y[n] can be calculated with same procedure. The resulting function

[ ] [ ] ∑

⁻

[ ]

=

1

0

1 ^K

k

j k XY j

XY j

XY S f

f K S f

CPSD with

[ ] [ ] [ ]

j

k j k j k

XY f X f Y f

S = ^*⋅ (14)

is a complex function and is called cross power spectral density (CPSD). It is the analogue to the cross correlation function in the frequency domain.

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The ratio

[ ] [ ]

[ ]

j XY

[ ]

j XX

j XY j

XY APSD f APSD f

f f CPSD

2

2 =

γ (15)

is called coherence function. The co-domain are real numbers between zero and one.

If the value is zero there is no common component with this frequency in the input signals. If the value is one, the whole signal at this frequency has a common source.

Common component/source means that both input signals have components with a common frequency and phase shift at this frequency, which do not change within the sequence used for the PSD calculation.

The phase function is the phase shift θ_xy as a function of frequency between coherent components of two different signals.

[ ] ( [ ] )

[ ]

(

XY j

)

j XY j

XY CPSD f

f f CPSD

Re arctanIm

θ = ₍₁₆₎

In many cases the use of estimators based on two sensor signals like the coherence and phase function improves the performance of the signal analysis by minimizing the influence of uncorrelated noise on the analysis.

Again, a special case is the harmonic analysis because of the special properties mentioned above. Also, the data are synchronized to a reference signal, so that the phase of the signal itself has a physical meaning and not only the phase shift relative to a second signal. Thus, in this case the phase function is estimated directly from the FFT with

[ ] ( [ ] )

[ ] (

j

)

j j

XX X f

f f X

Re arctanIm

θ = with 0< j< N (17)

where X is the Fourier transformation of x and N is the length of the window used for the discrete Fourier transformation.

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7.4. Coherent transfer functions

For constant-parameter linear systems the deformation of a signal y[tn] by physical processes could be expressed by a convolution of the input signal x[tn] with a transfer function h[tn]. A system has constant parameters if all fundamental properties of the system are invariant with respect to time. A system is linear if the response characteristics are additive and homogeneous.

In the frequency domain a convolution is a multiplication and so the transfer function H[fn] is the ratio of the output function Y[fn] to the input function X[fn].

[ ] [ ] [ ]

[ ]

^{[ ]} ^{[ ]}

( _Y f_n _X f_n )

i n n n

n

n e

f X

f Y f X

f f Y

H = = ^ϕ ⁻^ϕ (18)

Even so, in reality the output signal is contaminated with noise N[fn] that is independent from the input signal.

[ ]

^f_n ^H

[ ] [ ]

^f_n ^X ^f_n ^N

[ ]

^f_n

Y = ⋅ + (19)

An Estimator that minimizes the noise is called optimal estimator. In Bendat and Piersol (2000), the authors showed that the optimal estimator for the transfer function

[ ]

fn

Hˆ _is

[ ] [ ]

[ ]

_n

XX n XY

n APSD f

f f CPSD

Hˆ = (20)

and that its auto power spectral density is given by

[ ] [ ]

[ ] [ ] [ ]

[ ]

_n

XX n YY n

n XX

n XY H n

H APSD f

f f APSD

f APSD

f f CPSD

APSD = ₂ = ² ⋅

2 ˆ

ˆ γ (21)

The APSD of the transfer function shows at which frequency the signal is damped and where it is amplified. An estimator for the phase shift ϕ_Y

[ ]

f_n −ϕ_X

[ ]

f_n in formula (20) is the phase function θXY

[ ]

fj as defined in formula (17) because

[ ] ⁽ ^{[ ]} ⁾

[ ]

( )

[ ] [ ]

( )

[ ] [ ]

( ) ( ^{[ ]} )

( [ ]

n

)

n n

XX n

n XX n

n XY

n XY j

XY H f

f H f

APSD f

H

f APSD f

H f

CPSD f f CPSD

Re ˆ Im ˆ ˆ /

Re ˆ / Im Re

Im = =

θ = (22)

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7.5. Statistical Errors of coherence, power and phase spectra

Following Bendat and Piersol (2000) with formula 9.73 in their book the normalized random error of an estimator function is given by

[ ( ) ] ⁽ _{( )} ^{( )} ⁾ ^{( )}

( )

_d

XY XY vv

vv vv

n f

f f

G f f G

G

⋅

= −

= 2

2 2

ˆ ˆ

γ σ γ

ε (23)

with Gˆ_vv as estimator function of G_vv, the standard deviation σ

( )

Gˆvv and n_d as number of averages used for the calculation of the coherence function. Yet, the authors recommend that the coherence function as it is introduced in (15) itself is an estimator of a real coherence involving bias errors from a number of sources, just like the transfer function estimators. They summarize these errors as follows :

1.) Inherent bias in the estimation procedure 2.) Bias due to propagation time delays

3.) Nonlinear and/or time varying system parameters

4.) Bias in auto-spectral and cross-spectral density estimates 5.) Measurement noise at the input point

6.) Other inputs that are correlated with the measured input

There is no general way to avoid all these errors or to make a correction of the estimated values. Still, in most cases 1.) may be neglected if the random error from (23) is small. 2.) can be avoided if the window length used for the FFT is long compared to the traveling time of the observed phenomena. For example, the time of a pressure pattern traveling with the flow from one sensor to another should be small compared to the window length. If this is not the case the bias error might be corrected by

xy

xy G

Gˆ 1 T ˆ



 



 −

≈ τ

(24) with T as window length and τ as traveling time. For 3.) such kind of treatment does not exist. Here the problem has to be solved from the physics. This means that the bias error is smaller if the system fulfills the conditions of a constant parameter linear system. Point 4.) causes serious errors if two peaks are in one frequency slot. This could be avoided by an improved frequency resolution. Point 5.) and 6.) must be treated in the experiment by minimizing unwanted sources at the input point. If this is not possible but the sources are known, corrections might take place (see for example Bendat and Piersol (2000) for details).

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For the coherence itself, the relative random error is approximately given by

[ ( ) ]

( ) ( ( )

^f

)

n f

f _XY

d XY

xy 2

2

2 2 1

ˆ γ

γ γ

ε −

≈ ⋅ (25)

0.00 0.25 0.50 0.75 1.00

0.0 0.1 0.2 0.3 0.4 0.5

0 250 500 750 1000

1E-4 1E-3 0.01 0.1 1

n_d=500 n_d=250

n_d=1000 n_d=100 n_d=50

n_d=25

coherence γ²_xy

relative random error ε

nd=10

γ²

xy=0,5 γ²

xy=0,7 γ²_xy=0,8

γ²

xy=0,9

γ²

xy=0,95

γ²_xy=0,99 γ²

xy=0,25 γ²

xy=0,1

nd

FIG. 6 RELATIVE RANDOM ERROR OVER COHERENCE AND NUMBER OF AVERAGED WINDOWS

In Tab. 9.6 in reference Bendat and Piersol (2000) the normalized random error of common estimator functions are summarized. For power spectral density function and coherence the formulas are given by (23) and (25). For the transfer function it is given by :

( ( ) ) ^{( )}

( )

_d

XY xy XY

n f f f

H

⋅

= −₂

2

2 ˆ 1

γ

ε γ (26)

For the phase, the normalized random error can not be used, but the standard deviation in degrees is approximately given by: